A large part of mammalian brains is dedicated to sensory processing. Sensory receptors extract distinct but partly correlated features of the environment that vary at different timescales, and then transmit them through subcortical structures and a succession of recurrently connected cortical areas. A general theory on how sensory pathways encode, propagate, and transform sensory features is still missing. To understand how biophysically realistic neural networks support these fundamental brain functions, we developed an analytical framework based on the principle of efficient coding (Barlow, 1961, Olshausen and Field, 1997, Boerlin et al., 2013). Each brain area was conceptualized as a neural processing layer (hereafter, layer), and we assumed that the objective of each layer is to efficiently encode and transform multiple time-dependent stimulus features by optimally trading off the objectives of accurate information encoding and minimizing metabolic cost (Niven, 2016). Further, we assumed that the feedforward input to the next layer is a linear combination of multiple population-level readouts from the previous layer. We analytically derived a mechanistic model of neural dynamics that maximizes these coding objectives. Our optimal solution was a set of spiking networks with generalized leaky integrate-and-fire (LIF) neurons endowed with spike-triggered adaptation. Feedforward and recurrent synaptic connections were structured with like-to-like connectivity patterns. Recurrent connections realized efficient coding through lateral inhibition, while the like-to-like structure of feedforward connectivity supported reliable transmission of population signals across layers. By subjecting the feedforward connectivity to positive or negative interactions across sensory features, we created transformations in the representations that are synergistic or contrasting over time, respectively. Our framework is the first extension of efficient coding to multiple layers and provides groundwork for understanding neural computations for hierarchical sensory processing within biological brains.